Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for exams and because I was lead to it through a very convoluted path of research (from arithmetic through algebraic geometry to D-modules = holomorphic linear differential equations).
Anyway, I was told that PDEs were classified into 3 families: hyperbolic, elliptic, parabolic respectively. I saw examples for each: Wave, Laplace and Heat equations. I also saw lots of different methods to solve these examples: using symmetry to reduce to an ODE, Fourier transform, Distributions, Fourier series (in dimension 1 to deal with boundary conditions). Each time it seemed the answer was "reduce the problem to a polynomial equation or an ODE by any way you can". So my geometric brain kicked in and tried to give a unified geometric interpretation to all this.
I think I understand the distinction between those equations. We are considering 2nd order PDEs with real coefficients. And the idea is to reduce the classification to that of their principal symbols seen as quadratic forms on $\mathbb{R}^n$.
So I asked this kind a questions: let's consider a LINEAR PDE $Pf = 0$ where $P$ is a linear differential operator of degree $d$ on $\mathbb{R}^n$.
Question 1: Let's say a "naive elliptic PDE" is any PDE given by a differential operator $P$ whose principal symbol $\sigma(P)$ satisfies $\sigma(P)(x,\xi) \neq 0$ for $\xi\neq 0$. Is this definition any good?
If answer to question 1 is yes,
Question 2: What is the analogue of a parabolic or hyperbolic operator?
The obvious perfectly nice answer would be: "PDE's are classified by the hypersurfaces defined by their principal symbols". Unfortunately the answer I got was something like "Don't over-think it, the classification is more heuristic than anything.". Does that mean "there is a classification along theses lines but it is a bit more subtle" or that "things are much much more complicated in higher dimensions/degrees"?
Anyway…
Question 3: Is it at least true that the classification of PDE with constant coefficient is related to the classification of real algebraic projective hypersurfaces $\{\sigma(P) = 0 \} \subset \mathbb{P}^{n-1}_{\mathbb{R}}$?
Let's assume the previous questions aren't completely wrong for trivial reasons. Let's consider a PDE with non-constant coefficients. We should then classify those according to "families of algebraic projective hypersurfaces" in $\mathbb{P}(T^*\mathbb{R}^n)$.
Question 4: Which kind of families can we expect? Is that related to Gabber's theorem on involutivity of the characteristic variety?
I am now assuming someone answered all these questions without thinking the words "this is so completely wrong". I have a final question (at least before the next one):
Question 5: Why is this all so hard to learn/teach?
PS: Thanks to the people who took time to answer (lots of food for thoughts). I'd be happy to read more especially if you have some references.
Best Answer
PDE books often discuss classification, but they always restrict attention to the case of second order equations, especially for one function of several variables, with good reason. The point of a classification is to find categories of PDE whose analysis has many common features, but there really isn't any general classification in that sense, since the world of PDE is a huge zoo (once you leave the 3 familiar families of elliptic, hyperbolic and parabolic). Think about how you would define parabolic PDEs, even in second order. You already need to look beyond the symbol to distinguish $\partial_t u=\partial_{xx} u$ from $0=\partial_{xx} u$. As the OP points out, the symbol is certainly an important part of the ``classification''. The symbol is only a part of the tableau, which gives a little more information in an algebraic format; see the book of Bryant, et. al, Exterior Differential Systems. But systems of differential equations with the same tableau often have different analysis. Think about the famous Lewy counterexample. There are so many very different genera of animals in the zoo, and broad classifications don't give us much insight. Also look at Gromov, Partial Differential Relations, for lots of examples of PDEs that are locally the same, but globally very different, and are nothing like elliptic, hyperbolic or parabolic. So question 1: yes, question 2: hyperbolic is tricky to define beyond second order, because already for second order, hyperbolic is very different from ultrahyperbolic, so you really need something to distinguish a Lorentzian geometry from a more general pseudo-Riemannian geometry. On the other hand, your definition of ellipticity is perfect, and does give us some tools to carry out analysis. question 3: a little bit like yes, in that each PDE system gives rise to an algebraic variety, but finally no in that the classification of constant coefficient PDE systems is much finer than the classification of their symbols (it is in fact exactly the classification of their tableau), question 4: yes, you prolong until you hit involution, and so the classification of involutive tableau is not known, a huge messy algebra problem, question 5: like biology, it is messy because there are too many very different animals.